# Please help?

##### 2 Answers

36

#### Explanation:

When a line is a perpendicular bisector of another line, then that means that it divides the line into 2 equal parts. It also means that the lines are at right angles to each other

For example, if AC is a perpendicular bisector of DE, then DB=BE

If we know that DB=BE, AC is at right angles to DE and AB is a common side of

Why? By using congruent triangles

In

1. AB is the common side

2. DB=BE (AC is a bisector of DE)

3. AC is at right angles to DE (AC is a perpendicular bisector of DE)

then

Therefore, AD=AE (same sides of proven congruent triangles are equal)

Since we want to find side AE which is equal to

Since

- the lengths of
#bar(DB)# and#bar(BE)# are the same (since#bar(DE)# is divided in two equal parts), and angle#/_ABE# is#90^@# (since#bar(AC)# intersects#bar(DE)# perpendicularly). - triangle
#DeltaDBA# is a horizontal reflection of triangle#DeltaABE# (since#bar(AC)# divides one triangle#DeltaDAE# into two identical halves).

It then follows from the triangles being a reflection of each other that the lengths of

#3x - 9 = x + 21#

#=> 2x - 9 = 21#

#=> 2x = 30#

#=> x = 15#

As a result, the length of

#color(blue)(L) = x + 21 = 15 + 21 = color(blue)(36)#